PPT classifiable families of bipartite states

2010 ◽  
Vol 10 (5&6) ◽  
pp. 535-538
Author(s):  
F.E.S. Steinhoff ◽  
M.C. de Oliveira
Keyword(s):  
Numerical Analysis ◽  
Density Matrix ◽  
Block Structure ◽  
Arbitrary Dimension ◽  
Entangled States ◽  
Sufficient Criterion ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Bipartite States ◽  
Global Density

We construct a family of bipartite states of arbitrary dimension whose eigenvalues of the partially transposed matrix can be inferred directly from the block structure of the global density matrix. We identify from this several subfamilies in which the PPT criterion is both necessary and sufficient. A sufficient criterion of separability is obtained, which is fundamental for the discussion. We show how several examples of states known to be classifiable by the PPT criterion indeed belong to this general set. Possible uses of these states in numerical analysis of entanglement and in the search of PPT bound entangled states are briefly discussed.

scholarly journals Combinatorial laplacians and positivity under partial transpose

2008 ◽  
Vol 18 (1) ◽  
pp. 205-219 ◽  
Author(s):  
ROLAND HILDEBRAND ◽  
STEFANO MANCINI ◽  
SIMONE SEVERINI
Keyword(s):  
Density Matrix ◽  
Upper Bound ◽  
Block Structure ◽  
Alternative Proof ◽  
Density Matrices ◽  
Partial Transpose ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Degree Criterion

The density matrices of graphs are combinatorial laplacians normalised to have trace one (Braunstein et al. 2006b). If the vertices of a graph are arranged as an array, its density matrix carries a block structure with respect to which properties such as separability can be considered. We prove that the so-called degree-criterion, which was conjectured to be necessary and sufficient for the separability of density matrices of graphs, is equivalent to the PPT-criterion. As such, it is not sufficient for testing the separability of density matrices of graphs (we provide an explicit example). Nonetheless, we prove the sufficiency when one of the array dimensions has length two (see Wu (2006) for an alternative proof). Finally, we derive a rational upper bound on the concurrence of density matrices of graphs and show that this bound is exact for graphs on four vertices.

scholarly journals Separability of diagonal symmetric states: a quadratic conic optimization problem

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 45 ◽  
Author(s):  
Jordi Tura ◽  
Albert Aloy ◽  
Ruben Quesada ◽  
Maciej Lewenstein ◽  
Anna Sanpera
Keyword(s):  
Quantum Information ◽  
Optimization Problem ◽  
Entangled States ◽  
Conic Optimization ◽  
Entanglement Witnesses ◽  
Partial Transposition ◽  
Separability Problem ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Symmetric States

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS inCd⊗Cd(symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states ford≤4. Furthermore, ford≥5, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus onN-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.

EQUIVALENT SEQUENTIAL AND PARALLEL REDUCTIONS IN ARBITRARY BINARY PICTURES

2014 ◽  
Vol 28 (07) ◽  
pp. 1460009 ◽  
Author(s):  
KÁLMÁN PALÁGYI
Keyword(s):  
Single Point ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sufficient Criterion ◽  
Translation Invariant ◽  
Necessary And Sufficient

A reduction transforms a binary picture only by changing some black points to white ones, which is referred to as deletion. Sequential reductions traverse the black points of a picture, and consider a single point for possible deletion, while parallel reductions can delete a set of black points simultaneously. Two reductions are called equivalent if they produce the same result for each input picture. A deletion rule is said to be equivalent if it yields a pair of equivalent parallel and sequential reductions. This paper introduces a class of equivalent deletion rules that allows us to establish a new sufficient condition for topology-preserving parallel reductions in arbitrary binary pictures. In addition we present a method of verifying that a deletion rule given by matching templates is equivalent, a necessary and sufficient condition for order-independent deletion rules, and a sufficient criterion for order-independent and translation-invariant parallel subfield-based algorithms.

scholarly journals Quantum limit of genuine tripartite correlations by bipartite extremality

2020 ◽  
Vol 18 (04) ◽  
pp. 2050014
Author(s):  
Hiroyuki Ozeki ◽  
Satoshi Ishizaka
Keyword(s):  
Probability Space ◽  
Quantum Correlations ◽  
Quantum Limit ◽  
Sufficient Criterion ◽  
Extremal Points ◽  
Chsh Inequality ◽  
Nonlocal Correlations ◽  
Necessary And Sufficient

The characterization of the extremal points of the set of quantum correlations has attracted wide interest. In the simplest bipartite Bell scenario, a necessary and sufficient criterion for identifying extremal correlations has recently been conjectured, but extremality of tripartite correlations is not well known. In this study, we analyze tripartite extremal correlations in terms of the conjectured bipartite extremal criterion, and we demonstrate that the bipartite part of some extremal correlations satisfies the bipartite criterion, even though they violate Svetlichny’s inequality, and therefore are considered (stronger) genuine tripartite nonlocal correlations. This phenomenon arises from the fact that the conjectured extremal criterion is automatically satisfied when the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality exceeds a certain threshold, the value of which is given by the maximum CHSH violation at the edges of the probability space. This also suggests the possibility that the extremality of bipartite correlations can be certified by verifying whether the CHSH violation exceeds the threshold.

A note on the degree conjecture for separability of multipartite quantum states

2021 ◽  
pp. 2050048
Author(s):  
Zhen Wang ◽  
Ming-Jing Zhao ◽  
Zhi-Xi Wang
Keyword(s):  
Quantum States ◽  
Point Graph ◽  
Mixed States ◽  
Degree Condition ◽  
Graph Laplacians ◽  
Nearest Point ◽  
Necessary And Sufficient ◽  
Ppt Criterion ◽  
Math Phys ◽  
Degree Criterion

The degree conjecture for bipartite quantum states which are normalized graph Laplacians was first put forward by Braunstein et al. [Phys. Rev. A 73 (2006) 012320]. The degree criterion, which is equivalent to PPT criterion, is simpler and more efficient to detect the separability of quantum states associated with graphs. Hassan et al. settled the degree conjecture for the separability of multipartite quantum states in [J. Math. Phys. 49 (2008) 0121105]. It is proved that the conjecture is true for pure multipartite quantum states. However, the degree condition is only necessary for separability of a class of quantum mixed states. It does not apply to all mixed states. In this paper, we show that the degree conjecture holds for the mixed quantum states of nearest point graph. As a byproduct, the degree criterion is necessary and sufficient for multipartite separability of [Formula: see text]-qubit quantum states associated with graphs.

Indecomposability of entanglement witnesses

2015 ◽  
pp. 478-488
Author(s):  
Xiao-Fei Qi ◽  
Jin-Chuan Hou
Keyword(s):  
Entangled States ◽  
Finite Dimensional ◽  
Entanglement Witnesses ◽  
Ppt Criterion

We present a way to construct indecomposable entanglement witnesses from any permutations pi with pi^2 not equal to id for any finite dimensional bipartite systems. Some new bounded entangled states are also found, which can be detected by such witnesses while cannot be distinguished by PPT criterion, realignment criterion, etc.

scholarly journals An Exactly Solved Model for Mutation, Recombination and Selection

2003 ◽  
Vol 55 (1) ◽  
pp. 3-41 ◽  
Author(s):  
Michael Baake ◽  
Ellen Baake
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Sufficient Criterion ◽  
Linkage Disequilibria ◽  
Additive Type ◽  
Mean Fitness ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Independent Mutation ◽  
Selection Of

AbstractIt is well known that rather generalmutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

Output Feedback Assignability for Nonlinear Min-Max Systems

2011 ◽  
Vol 314-316 ◽  
pp. 374-379
Author(s):  
Hong Yun Wei ◽  
Zhong Xun Zhu ◽  
Yue Gang Tao ◽  
Wen De Chen
Keyword(s):  
Output Feedback ◽  
Cycle Time ◽  
Closed Loop ◽  
Loop System ◽  
Closed Loop System ◽  
Sufficient Criterion ◽  
Feedback Cycle ◽  
Necessary And Sufficient ◽  
Coloring Graph

This paper investigates the output feedback cycle time assignability of the min-max systems which are more complex than the systems studied in recent years. Max-plus projection representation for the closed-loop system with min-max output feedback is introduced. The coloring graph is presented and applied to analyze the structure of systems effectively. The necessary and sufficient criterion for the output feedback cycle time assignability is established which is an extension of the results studied before. The methods are constructive in nature.

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